Existence and Uniqueness of Positive Solutions for Singular Nonlinear Fractional Differential Equation via Mixed Monotone Operator Method

Abstract

In this article, we discuss the existence and uniqueness of positive solution for a class of singular fractional differential equations, where the nonlinear term contains fractional derivative and an operator. By applying the fixed point theorem in cone, we get the existence and uniqueness of positive solutions for the fractional differential equation. Moreover, we give an example to demonstrate our main result.

1. Introduction

In this article, we consider the existence and uniqueness of solutions for the following singular fractional differential equations: where , , , , and are allowed to be singular at or , is continuous and may be singular at and , is continuous and may be singular at , and is continuous function.

In recent years, we can see that differential equations have more and more attention in many fields, like electrical networks, quantum physics, and probability. There are a lot of interesting results from fractional differential equation, Schrodinger equation and k-hessian equations. In [1–4], the authors studied the solutions of fractional differential equations, and obtained some interesting results. Such as uniqueness of iterative positive solution, existence of multiple positive solutions or maximum and minimum solutions. In [5–9], the authors studied the solutions of Schrödinger equation. From these paper, there have several results about existence of infinitely solutions, existence and nonexistence of blow-up solutions, existence and asymptotic properties of solutions or existence and nonexistence of entire large solutions. In [10–12], the authors considered existence of blow-up radial solutions of the k-hessian equations or convergence analysis and uniqueness of blow-up solutions of the k-hessian equations. Inspired by these conclusions,we realize that the questions about differential equations are very interesting and varied. So we focus on fractional differential equation. The authors in [13] studied the following fractional differential equation: where , , is the Riemann-Liouville fractional derivative, and is a continuous function.

In [14], by using cone expansion fixed point theorem, Li et al. studied the existence of positive solutions for the following fractional differential equation with Riemann-Stieltjes integral: where , , are continuous, , and allowed to be singular at or , are continuous and may be singular at , is continuous with , where is the Riemann-Stieltjes integral with a signed measure and is a bounded variation function, and is the standard Riemann-Liouville derivative of order .

In [15], Liu et al. investigated the iterative positive solutions for the following singular nonlinear fractional differential equation with integral boundary conditions. where , , are continuous and allowed to be singular at or , is continuous and may be singular at and , is continuous and may be singular at and , is continuous with , where is the Riemann-Stieltjes integral with a signed measure and is a bounded variation function, and is the standard Riemann-Liouville derivative of order .

In [16], Zhang and Tian considered the following fractional differential equation: where , , , and , , and are continuous functions.

Motivated by the abovementioned papers, the purpose of this article is to study the existence and uniqueness of positive solution for FBVP (1). Obviously, the problem in our article is more general. We not only consider the nonlinear term containing the derivative term, but also study the nonlinear term containing an operator. And we do not need the linearity of operator in nonlinear term. It may be a nonlinear operator.

Comparing with the results in [17], boundary value problem (1) has a more general form. First, we discuss the singularity. It means that are allowed to be singular at , may be singular at and , and may be singular at . Second, we not only consider the derivative term but also consider the operator term, where the operator can be linear or nonlinear. Especially, when and , we can see that the problem in [16] is a special case of problem (1). Comparing with [13], firstly, when , , , and , then our problem (1) reduced compared to the problem in [13]. Third, in [13], the author did not consider the singularity, but in problem (1), we consider the singularity. Compared with [14], first, our method is different from that of [14]. We use the fixed point theorem for the sum of operators rather than the fixed point theorem of cone expansion and compression. Second, not only has three variables but also is singular for both time and space variables, and also has three variables. Compared with [15], we consider the derivative term and the operator term. Our results extended and improved some existing results, such as [13–16].

The rest of the paper is organized as follows: in Section 2, we give some preliminaries and lemmas that will be used in our main result. In Section 3, we study the existence and uniqueness of a positive solution of the BVP (1). In Section 4, an example is provided to demonstrate our main results.

2. Preliminaries and Lemmas

In this section, we present some notations, definitions and lemmas that will be used in the paper.

Let be a Banach space, and is defined as the zero element of . A nonempty closed convex set is a cone if it satisfies (1) and (2) . Putting , a cone is said to be solid if is nonempty. In this paper, suppose that is a Banach space partially ordered by a cone , that is, if and only if . A cone called normal if there exists a constant such that for all implies , and the smallest such is called the normality constant of . For all , we denote the notation if there have constants such that . Clearly, is an equivalence relation. Given , we denote by the set . And it is easy to know that .

Definition 1 (see [18]). An operator is said to be a mixed monotone operator if is increasing in the first component and decreasing in the second component, i.e., , implies . An element is called a fixed point of if .

Definition 2 (see [19]). Let or and be a real number with . An operator is said to be concave if it satisfies

Definition 3 (see [19]). An operator is said to be subhomogeneous if it satisfies

Definition 4 (see [20]). The Riemann-Liouville fractional integral of order of a function is given by provided that the right-hand side is pointwise defined on .

Definition 5 (see [20]). The Riemann-Liouville fractional derivative of order of a continuous function is given by where , where denotes the integer part of the number , provided that the right-hand side is pointwise defined on .

Lemma 6 (see [20]). Let . If we assume , then the fractional differential equation has as the unique solution, where .

From the definition of the Riemann-Liouville derivative, we can obtain the statement.

Lemma 7 (see [20]). Assume that with a fractional derivative of order that belongs to . Then, for some , where .

In the following, we present the Green’s function of FBVP (1).

Lemma 8 (see [16]). Let , then the unique solution of the linear problem is given by where

Lemma 9 (see [16]). The Green’s function defined by (15) satisfies the following properties: (1) is continuous(2)For any , there is (3)For any , there is

In [17], the operator equation was studied, where are mixed monotone operators, and for all , there have , . Based on the result in [17], we get the following lemma:

Lemma 10. Let be a normal cone in , and let be two mixed monotone operators and is a decreasing operator, suppose that (A1)for all , there have such that (A2)for all , (A3)for all , (A4)there exists and , such that (A5)there exists a constants , such that for all , Then (1)(2)There exist and such that (3)The operator equation has a unique solution in (4)For any initial values , constructing successively the sequences we have and as

Proof. Now we define the operator by

So it is easy to know that is a mixed monotone operator. Next, we will prove that and , where , Then according to the results of [21], the conclusions of Lemma 10 holds.

Firstly, we will show that for and .

From (18)–(20), for any , , there have

According to (A4), we may assume that there exist constants such that

Choose sufficiently small number such that for any ; then from (18)–(20) and (25)–(28), there have and we know . So and . Similarly, we can easily get that , , , and . Then, we have .

Second, we will prove that is satisfied for any .

For any , from (A5), we have

Then . So

Thus, there have

So

Now we note ; it is easy to know that , where . Then, .

Thus, by the results of Zhai and Zhang in [21], we know what the conclusions of Lemma 10 hold.

3. Main Results

For convenience, in this section, we set

Clearly, is a Banach space with the norm

Now, we let the cone be defined by

There have , and is a normal cone. And, the order relation in is given by if .

From Lemma 8, we get that the unique solution of the problem (1) satisfies the following equation: where is the Green’s function given in (15).

Theorem 11. Assume that the following assumptions hold: (H1) are continuous, and are allowed to be singular at or . For all , there is , where is a constant(H2) are continuous, may be singular at , and , may be singular at (H3)For fixed , is increasing in and decreasing in ; for fixed , is increasing in and decreasing in ; for fixed , is decreasing in with (H4)There exists a constant such that for all and , (H5). There exists a constant such that for all and , (H6) and (H7). and is increasing in . For and , there is Then problem (1) has a unique positive solution , where . For any initial value , there are two iterative sequences for approximating , that is, , as , where

Proof. We define operators as follows:

Furthermore, for , we obtain that

Clearly, is the solution of problem (1) if and only if . So, if we can prove that the operators satisfy all conditions in Lemma 10, then we will obtain the conclusions in Theorem 11.

First, we will prove that are well defined. Evidently, is well defined, so we omit it.

From (H4), for all , we have

Furthermore, we have

Taking in (H4) and (47), for , we have

For any , there exists a constant such that , where . So, from the above inequalities, we obtain

Then, combining assumption (H6), we get

Thus, we know that are well defined.

Second, we will prove that satisfy conditions (A1)–(A3) of Lemma 10. From (H1)–(H3), for all , we have . Therefore, . And for all , , means . Thus, from assumptions (H2) and (H3), it is clear that operators satisfy conditions (A1)–(A3) of Lemma 10. And are mixed monotone operators and is a decreasing operator.

Third, we show that . From (H2) and (H3) and Lemma 9, we have